Self-intersection local times of random walks: exponential moments in subcritical dimensions

Fix p > 1, not necessarily integer, with p(d - 2) < d. We study the p-fold self-intersection local time of a simple random walk on the lattice up to time t. This is the p-norm of the vector of the walker's local times, a"" (t) . We derive precise logarithmic asymptotics of the expectation of exp{theta (t) ||a"" (t) || (p) } for scales theta (t) > 0 that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of t and theta (t) , and the precise rate is characterized in terms of a variational formula, which is in close connection to the Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for ||a"" (t) || (p) /(tr (t) ) for deviation functions r (t) satisfying . Informally, it turns out that the random walk homogeneously squeezes in a t-dependent box with diameter of order a parts per thousand(a) t (1/d) to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.